An extended Flaherty-Keller formula for an elastic composite with densely packed convex inclusions

Abstract

In this paper, we are concerned with the effective elastic property of a two-phase high-contrast periodic composite with densely packed inclusions. The equations of linear elasticity are assumed. We first give a novel proof of the Flaherty-Keller formula for elliptic inclusions, which improves a recent result of Kang and Yu (Calc Var Partial Differ Equ 59(1):3, 2020). We construct a family of auxiliary functions consisting of the Keller-type functions and additional correctors which depend on the coefficients of Lamé system and the geometry of inclusions, to capture the full singular term of the gradient. On the other hand, this method allows us to deal with the inclusions of arbitrary shape, even with zero curvature. An extended Flaherty-Keller formula is proved for m-convex inclusions, $$m>2$$ , curvilinear squares with round off angles, which minimize the elastic energy under the same volume fraction of hard inclusions.